Classification problems in noncommutative algebraic geometry and representation theory

2011 ◽  
pp. 355-406 ◽  
Author(s):  
Izuru Mori
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Classification Problems ◽  
Noncommutative Algebraic Geometry

scholarly journals A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

2009 ◽  
Vol 52 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Jakob Cimprič
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Representation Theorem ◽  
Commutative Rings ◽  
Real Algebraic Geometry ◽  
New Approach ◽  
Noncommutative Version ◽  
Quadratic Modules ◽  
Rings With Involution ◽  
Associative Rings

AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

scholarly journals Relative Calabi–Yau structures

2019 ◽  
Vol 155 (2) ◽  
pp. 372-412 ◽  
Author(s):  
Christopher Brav ◽  
Tobias Dyckerhoff
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Riemann Surfaces ◽  
Fukaya Categories ◽  
Composition Law

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

scholarly journals VANISHING OF THE NEGATIVE HOMOTOPY -THEORY OF QUOTIENT SINGULARITIES

2017 ◽  
Vol 18 (3) ◽  
pp. 619-627
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Algebraic Geometry ◽  
Upper Bound ◽  
Homotopy Theory ◽  
Krull Dimension ◽  
Noncommutative Algebraic Geometry ◽  
Noetherian Scheme ◽  
Quotient Singularities ◽  
Cyclic Quotient Singularities

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy$K$-theory groups of a Noetherian scheme$X$of Krull dimension$d$vanish below$-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy$K$-theory groups vanish below$-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy$K$-theory group.

scholarly journals A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC

2020 ◽  
pp. 1-19
Author(s):  
Masaki Matsuno
Keyword(s):  
Algebraic Geometry ◽  
Complete Classification ◽  
Complete List ◽  
Graded Algebra ◽  
Regular Algebra ◽  
3 Dimensional ◽  
Noncommutative Algebraic Geometry ◽  
Regular Algebras ◽  
Point Scheme

Abstract Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$ -dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$ . In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$ -dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

scholarly journals Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

2021 ◽  
Vol 39 (1) ◽  
Author(s):  
Héctor Suárez ◽  
Duban Cáceres ◽  
Armando Reyes
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Noncommutative Differential Geometry ◽  
Regular Algebras ◽  
Skew Pbw Extensions ◽  
Finitely Presented

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its P-determinant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.

scholarly journals THE MONOID OF FAMILIES OF QUIVER REPRESENTATIONS

2002 ◽  
Vol 84 (3) ◽  
pp. 663-685 ◽  
Author(s):  
MARCUS REINEKE
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Representation Theory ◽  
Quantum Group ◽  
Subject Classification ◽  
Quiver Representations ◽  
Enveloping Algebras ◽  
Geometric Methods ◽  
Quantized Enveloping Algebras ◽  
Monoid Structure

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.

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